First order spectral perturbation theory of square singular matrix polynomials (Q846318)

Let \(\mathbb{C}^{1\times n}(\lambda)\) and \(\mathbb{C}^{n\times 1}(\lambda)\) denote the vector spaces of \(n\)-tuples of complex rational functions over the field of such functions, with elements considered as row, and respectively column vectors. Let \(P(\lambda)\) be an \(n\times n\) matrix polynomial of degree~\(l\) (i.e., \(P(\lambda)=A_0+\lambda A_1+\dots+\lambda^lA_l\) where \(A_0,\dots,A_l\in\mathbb{C}^{n\times n}, A_l\neq 0\)), and denote~\({\mathcal N}_T(P)=\{y(\lambda)\in\mathbb{C}^{1\times n}(\lambda):y(\lambda)P(\lambda)\equiv 0\},\,{\mathcal N}(P)=\{x(\lambda)\in\mathbb{C}^{n\times 1}(\lambda):P(\lambda)x(\lambda)\equiv 0\}\). The authors' main result (Theorem~3) is the following: Given a finite semisimple eigenvalue~\(\lambda_0\) of~\(P(\lambda)\) with geometric multiplicity~\(g\), let the rows of~\(W\in\mathbb{C}^{n\times n}\) and the columns of~\(V\in\mathbb{C}^{n\times n}\) be a basis of~\({\mathcal N}_T(P(\lambda_0))\) and \({\mathcal N}(P(\lambda_0))\) respectively. In the set of perturbations~\(M(\lambda)\) that are \(n\times n\) matrix polynomials with degree at most~\(l\), the pencil (1)~\(WM(\lambda_0)V+\zeta WP'(\lambda_0)V\) is generically regular and has exactly \(g\) finite eigenvalues (i.e., this holds for all~\(M(\lambda)\) except those in an algebraic manifold of positive codimension). Furthermore, if the pencil~(1) is regular and has exactly~\(g\) finite eigenvalues, denoted by \(\zeta_1,\dots,\zeta_g\), then there are exactly~\(g\) eigenvalues of~\(P(\lambda)+\varepsilon M(\lambda)\) such that \(\lambda_j(\varepsilon)=\lambda_0+\zeta_j\varepsilon+ o(\varepsilon),\,j=1,\dots, g\), as \(\varepsilon\to 0\). If \(g=1\) (i.e., \(\lambda_0\) is simple), then \(o(\varepsilon)\) can be replaced with~\(O(\varepsilon^2)\). The authors and \textit{J.~Moro} have previously studied the case~\(l=1\) [Linear Algebra Appl.~429, 548--576 (2008; Zbl 1154.15012)].